Michigan Mathematical Journal

Hyperbolicity Notions for Varieties Defined over a Non-Archimedean Field

R. Rodríguez Vázquez

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Abstract

Firstly, we pursue the work of W. Cherry on the analogue of the Kobayashi semidistance dCK, which he introduced for analytic spaces defined over a non-Archimedean metrized field k. We prove various characterizations of smooth projective varieties for which dCK is an actual distance.

Secondly, we explore several notions of hyperbolicity for a smooth algebraic curve X defined over k. We prove a non-Archimedean analogue of the equivalence between having a negative Euler characteristic and the normality of certain families of analytic maps taking values in X.

Article information

Source
Michigan Math. J., Volume 69, Issue 1 (2020), 41-78.

Dates
Received: 3 January 2018
Revised: 29 August 2018
First available in Project Euclid: 21 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1574326880

Digital Object Identifier
doi:10.1307/mmj/1574326880

Mathematical Reviews number (MathSciNet)
MR4071345

Zentralblatt MATH identifier
07208925

Subjects
Primary: 32P05: Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem)
Secondary: 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions

Citation

Rodríguez Vázquez, R. Hyperbolicity Notions for Varieties Defined over a Non-Archimedean Field. Michigan Math. J. 69 (2020), no. 1, 41--78. doi:10.1307/mmj/1574326880. https://projecteuclid.org/euclid.mmj/1574326880


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References

  • [ACW08] T. T. H. An, W. Cherry, and J. T.-Y. Wang, Algebraic degeneracy of non-Archimedean analytic maps, Indag. Math. (N.S.) 19 (2008), no. 3, 481–492.
  • [BR10] M. Baker and R. Rumely, Potential theory and dynamics on the Berkovich projective line, Math. Surveys Monogr., 159, American Mathematical Society, Providence, RI, 2010.
  • [Ber90] V. G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys Monogr., 33, American Mathematical Society, Providence, RI, 1990.
  • [Ber94] V. G. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Publ. Math. Inst. Hautes Études Sci. 78 (1994), 5–161, 1993.
  • [Ber06] F. Berteloot, Méthodes de changement d’échelles en analyse complexe, Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 3, 427–483.
  • [Bos77] S. Bosch, Eine bemerkenswerte Eigenschaft der formellen Fasern affinoider Räume, Math. Ann. 229 (1977), no. 1, 25–45.
  • [Bro78] R. Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213–219.
  • [Che93] W. Cherry, Hyperbolic $p$-adic analytic spaces, Thesis, Yale University, 1993.
  • [Che94] W. Cherry, Non-Archimedean analytic curves in Abelian varieties, Math. Ann. 300 (1994), no. 3, 393–404.
  • [Che96] W. Cherry, A non-Archimedean analogue of the Kobayashi semi-distance and its non-degeneracy on Abelian varieties, Illinois J. Math. 40 (1996), no. 1, 123–140.
  • [CTT16] A. Cohen, M. Temkin, and D. Trushin, Morphisms of Berkovich curves and the different function, Adv. Math. 303 (2016), 800–858.
  • [DR] S. Diverio and E. Rousseau, A survey on hyperbolicity of projective hypersurfaces, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2011. [On the title page: A survey on hyperbolicity of projective hypersurfaces].
  • [Duc14] A. Ducros, La structure des courbes analytiques, preprint, 2014, http://webusers.imj-prg.fr/antoine.ducros/livre.html.
  • [FKT12] C. Favre, J. Kiwi, and E. Trucco, A non-Archimedean Montel’s theorem, Compos. Math. 148 (2012), no. 3, 966–990.
  • [Gri15] N. Grieve, Diophantine approximation constants for varieties over function fields, Michigan Math. J. 67 (2018), no. 2, 371–404.
  • [JV18] A. Javanpeykar and A. Vezzani, Non-Archimedean hyperbolicity and applications, preprint, 2018.
  • [Kob67] S. Kobayashi, Invariant distances on complex manifolds and holomorphic mappings, J. Math. Soc. Japan 19 (1967), 460–480.
  • [Kob98] S. Kobayashi, Hyperbolic complex spaces, Grundlehren Math. Wiss., 318, Springer-Verlag, Berlin, 1998.
  • [KO75] S. Kobayashi and T. Ochiai, Meromorphic mappings onto compact complex spaces of general type, Invent. Math. 31 (1975), no. 1, 7–16.
  • [Lan87] S. Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, 1987.
  • [Nog92] J. Noguchi, Meromorphic mappings into compact hyperbolic complex spaces and geometric Diophantine problems, Internat. J. Math. 3 (1992), no. 2, 277–289.
  • [Pet09] C. Petsche, Nonarchimedean equidistribution on elliptic curves with global applications, Pacific J. Math. 242 (2009), no. 2, 345–375.
  • [Poi13] J. Poineau, Les espaces de Berkovich sont angéliques, Bull. Soc. Math. France 141 (2013), no. 2, 267–297.
  • [Poi14] J. Poineau, Sur les composantes connexes d’une famille d’espaces analytiques $p$-adiques, Forum Math. Sigma 2 (2014), e14.
  • [Rod16] R. Rodríguez Vázquez, Non-Archimedean normal families, preprint, 2016.
  • [Roy71] H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970), Lecture Notes in Math., 185, pp. 125–137, Springer, Berlin, 1971.
  • [Sam66] P. Samuel, Lectures on old and new results on algebraic curves, Notes by S. Anantharaman. Tata Institute of Fundamental Research Lectures on Mathematics, 36, Tata Institute of Fundamental Research, Bombay, 1966.
  • [Tem15] M. Temkin, Introduction to Berkovich analytic spaces, Berkovich spaces and applications, Lecture Notes in Math., 2119, pp. 3–66, Springer, Cham, 2015.
  • [Thu05] A. Thuillier, Potential theory on curves in non-Archimedean geometry. Applications to Arakelov theory, Thesis, Université Rennes, 1, 2005.
  • [Tsu79] R. Tsushima, Rational maps to varieties of hyperbolic type, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 95–100.
  • [Voi03] C. Voisin, On some problems of Kobayashi and Lang; algebraic approaches, Current developments in mathematics, 2003, pp. 53–125, Int. Press, Somerville, MA, 2003.
  • [Zal75] L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813–817.